A light rendering problem or an export problem?
I think the issue is that the OBJ file format does not support any type of lighting information. It simply saves geometry, simple materials, and links to image textures (via the .mtl file). 

I think perhaps your first example with the white lights is misleading you, as I don't think you're actually seeing the surfaces lit by the lights in the other 3d app, you're just seeing the image textures shown on the surfaces?